Number & Operations for Teachers 

Copyright David & Cynthia Thomas, 2009

Greater Than - Less Than

When comparing two base ten numbers having the same number of digits, such as 289 and 301, the question of which number is greater is answered by comparing the numerals, place-by-place, beginning with the left-most place. Ignoring the numerals in the tens and ones places of each number, the decision depends only on the fact that, in one number, there is a 3 in the hundreds place while, in the other number, there is a 2 in the hundreds place. This fact alone guarantees that 301 > 289. Why is this true? What argument could you make using base ten blocks to support this procedure? And why are the numerals in the tens and ones place irrelevant in this case? Under what circumstances would the numerals in the tens and ones places be significant?

It is instructive to discuss these questions with other students, for different people often have different ideas about how to explain why 301 > 289. For instance, one individual might explain that the number 301 may be thought of as 30 tens plus 1 one, while 289 may be represented as 28 tens plus 9 ones. Since 30 tens is clearly greater than 28 tens, 301 > 289. Another person might use base ten blocks to model these numbers and explain why one number is greater than the other. By listening to one another, students enrich one another’s understanding. As teachers of mathematics, we must be prepared to offer multiple representations to all of our students and allow them to decide which representation makes the most sense.

Learning mathematics must involve more than memorizing and applying rules correctly. It should also include an ever-deepening understanding of their meaning, their representations, and their applications. The following section presents on a conceptual level a few of the implications of this statement.